Example Question - combine fractions
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Simplifying an Algebraic Expression
The image shows an algebraic expression that needs to be simplified: \[ \frac{1 - \sqrt{2}}{\sqrt{5} - \sqrt{3}} + \frac{1 + \sqrt{2}}{\sqrt{5} + \sqrt{3}} \] To simplify this expression, we can combine the two fractions into a single fraction by finding a common denominator, which is \((\sqrt{5} - \sqrt{3})(\sqrt{5} + \sqrt{3})\). First, we recognize that \((\sqrt{5} - \sqrt{3})(\sqrt{5} + \sqrt{3})\) equals \(\sqrt{5}^2 - \sqrt{3}^2\), which simplifies to \(5 - 3\), or \(2\). Now, let's work out the combined numerator: For denominator \(\sqrt{5} - \sqrt{3}\), the corresponding numerator is \((1 - \sqrt{2})(\sqrt{5} + \sqrt{3})\). For denominator \(\sqrt{5} + \sqrt{3}\), the corresponding numerator is \((1 + \sqrt{2})(\sqrt{5} - \sqrt{3})\). Let's find each product for the numerators separately: - \((1 - \sqrt{2})(\sqrt{5} + \sqrt{3}) = \sqrt{5} + \sqrt{3} - \sqrt{10} - \sqrt{6}\) - \((1 + \sqrt{2})(\sqrt{5} - \sqrt{3}) = \sqrt{5} - \sqrt{3} + \sqrt{10} - \sqrt{6}\) Next, combine these to get the new numerator: \[ (\sqrt{5} + \sqrt{3} - \sqrt{10} - \sqrt{6}) + (\sqrt{5} - \sqrt{3} + \sqrt{10} - \sqrt{6}) \] Simplify by cancelling out the opposite terms \( \sqrt{3} \) and \( -\sqrt{3} \), and by combining like terms: \[ 2\sqrt{5} - 2\sqrt{6} \] With the denominator found to be \(2\), we now have: \[ \frac{2\sqrt{5} - 2\sqrt{6}}{2} \] Divide both terms in the numerator by the denominator to simplify further: \[ \sqrt{5} - \sqrt{6} \] Thus, the simplified expression is: \[ \sqrt{5} - \sqrt{6} \]
Simplifying an Algebraic Expression
To simplify the given algebraic expression: \[\frac{\frac{2}{x} - 5}{6 + \frac{3}{x}}\] First, find a common denominator for the fractions in the numerator and the denominator. The common denominator for the fractions with \(x\) in the numerator is \(x\), and similarly for the denominator. Rewrite the expression with common denominators as follows: \[\frac{\left(\frac{2}{x} - \frac{5x}{x}\right)}{\left(6\frac{x}{x} + \frac{3}{x}\right)}\] This simplifies to: \[\frac{\frac{2 - 5x}{x}}{\frac{6x + 3}{x}}\] Now, since the denominators are the same, the fractions can be combined: \[\frac{2 - 5x}{6x + 3}\] This is the simplified form of the given expression. It cannot be further simplified as there are no common factors between the numerator and the denominator that can be cancelled out.
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